Analysis of Linear and Nonlinear Quasiperiodic Metamaterials with Local Resonators

Abstract

This dissertation investigates the linear and nonlinear topological phenomena of quasiperiodic metamaterials with the inclusion of local resonators using both analytical and numerical approaches. The proposed metamaterial is modeled as a spring-mass chain with each mass coupled to a single spring-mass resonator. In the first part of this work, the topological edge modes are utilized to improve energy harvesting capabilities via the inclusion of electromechanical resonators. The resonators are modeled as a lumped parameter piezoelectric material shunted to a single resistor. Through eigenvalue and Chern number analyses, the effect of the electromechanical coupling on the band structure and topological nature of the system is observed. The results indicate that weak electromechanical coupling has no impact on the band structure or topological phenomena. Furthermore, the localized edge modes provide significant improvements for energy harvesting applications.

The next part of this dissertation reveals how local resonators and quasiperiodic patterning work individually and cooperatively to form bandgaps. To achieve this, quasiperiodic patterning is introduced to resonator parameters as well as the main cell stiffness. Multiple analytical and numerical methods of analysis are used. Eigenvalue analysis is used to obtain an analytical dispersion relation for each case, and the inverse method is used to characterize the formation mechanism for each bandgap. Chern number and integrated density of states analysis is used to determine the topological nature of each bandgap. Results indicate that, by introducing patterning to the resonator parameters, the band structure becomes significantly more tunable, and many unique phenomena are produced. Multiple locally resonant bandgaps are created, which provide stronger vibration attenuation and display coupled topological and locally resonant behavior. Finally, a thorough parametric study is performed for each case to guide designers in selecting ideal parameters and pattering for a variety of applications.

The remainder of this dissertation explores how these locally resonant topological features behave when expanded into the nonlinear regime. To model this, multiple cases of nonlinearity are considered, with nonlinearity present in the main cell stiffness, the resonator stiffness, and both main cell and resonator stiffnesses. Multiple perturbation methods are used including the method of multiple scales and the harmonic balance method. Direct numerical integration is further used to validate the results. The band structure and mode shapes are both determined as a function of the excitation amplitude, with the results indicating a strong amplitude dependence for both. As the amplitude is increased, all modes experience a shift in frequency, which can result in loss of localization for some edge modes and the generation of breathers for some other modes. For a nonlinear chain, these nonlinear effects are limited to higher frequencies, but for nonlinear resonators, the effects are strongest at frequencies near to the resonator frequency. By including nonlinearity in both, it is possible to achieve a wide and tunable frequency range for amplitude-dependent localized solutions. This can be used for improved wave guiding, nonreciprocity, and amplitude filtering.